Statistics of Binary & Multiple StarsThe Astrophysical Journal Supplement Series, 230:15 (55pp),...

Statistics of Binary & Multiple Stars:Implications for Formation & Evolution

Max Moe (University of Arizona)The Impact of Binaries on Stellar Evolution

ESO Garching – July 3, 2017

Topics:§ Detection methods and selection effects§ Corrected joint pdf f(M1, P, q, e) ≠ f(M1) × f(P) × f(q) × f(e)

for systems near ZAMS (𝜏 ~ 5 Myr) and Z = Z☉§ Variations as a function of 𝜏, Z, and environment§ Implications for binary star formation and evolution

Main Resources:§ Review by Duchene & Kraus (2013)§ Meta-analysis by Moe & Di Stefano (2017)

Mind your Ps and Qs: f(P, q) ≠ f(P) × f(q) No. 1, 2010 A SURVEY OF STELLAR FAMILIES 27

Figure 11. Detection limits of the various companion search methods by period and mass ratio. For convenience, the periods are converted to semi-major axes for ouraverage mass sum of 1.5 M⊙ and shown on the top, and the mass ratios are converted to approximate ∆V values for primaries G0–K0 and shown on the right. The solidvertical line at log P just above 3.5 (10 years) is the limit for spectroscopic companions from the CCPS efforts, which are capable of detecting any companions to the leftof it. The solid curve turning vertical at about log P = 4 (30 years) marks the limit of the D. W. Latham et al. (2010, in preparation) spectroscopic survey. The dashedlines show the detection region of the CHARA SFP search, the dotted lines mark the boundary of speckle interferometric searches, the dash-dotted lines demarcatethe AO detection space, and the dash-triple-dotted lines delineate the CPM search region. The assumptions for detection limits for each technique are outlined inthe text. The overplotted points show the actual pairs found with estimates of periods and component masses. Plus signs identify spectroscopic orbits, crosses markvisual orbits, open diamonds show combined spectroscopic-visual solutions, filled circles identify CPM pairs and open squares show unresolved companions such aseclipsing binaries or overluminous sources.

2006; Allen et al. 2007; Lepine & Bongiorno 2007; Looperet al. 2007; Metchev & Hillenbrand 2009). While many of thesesearches specifically targeted brown dwarf companions, theywere more sensitive to, and often reported, the discovery of low-mass stellar companions as well. Additionally, a specific searchfor low-mass companions out to separations of 10,000 AUaround solar-type stars within 10 pc using 2MASS infrared im-ages resulted in no new detections (D. Looper 2009, private com-munication). Consistent with these findings, our results includeonly eight brown dwarf companions, found by high-resolutiontechniques such as coronagraphy, AO, and space-based obser-vations (e.g., Potter et al. 2002; Bouy et al. 2003; Burgasseret al. 2005; Luhman et al. 2007). Moreover, while our CPMefforts using archival images are not sensitive to brown dwarfs,widely separated substellar companions to nearby stars haveeffectively been identified using 2MASS infrared images (Kirk-patrick et al. 2001; Lowrance et al. 2002; Looper et al. 2007).While these efforts readily present new detections, null resultsoften remain unpublished. Nevertheless, these searches using2MASS infrared images have largely yielded null detections(D. Kirkpatrick 2009, private communication), suggesting a realshortage of low-mass companions. As noted above, the CCPSsurvey is specifically suited to identifying relatively long-periodlow-mass companions, or those in nearly face-on orbits, as low-amplitude temporal drifts in the radial-velocity measurementsof the host stars. Our search of the CCPS data revealed 21 starswhich have such drifts with rms deviation from a uniform ve-locity of less than 0.1 km s−1 over some 10 years. Fourteen of

these have known stellar companions, many of which are closeenough to cause the radial-velocity variations seen. The remain-ing seven stars do not have any known companions, implyingundetected stellar or sub-stellar companions to presumed sin-gle stars. While some of these could turn out to be planets inlong-period orbits, they could all be new stellar or brown dwarfcompanions. Added to the one missing CPM companion andtwo missing spectroscopic companions discussed above, we es-timate that our survey might be missing about 10 companionsto presumed single stars.

The extrapolation of new companions used above to estimatethe number of companions missed by our survey is validonly if the stars not observed by a particular technique have,in general, been historically covered in equal measure asthose observed by that technique. If the stars not surveyedby a given technique fall into a set of generally neglectedstars, the approach used above would underestimate missingcompanions. We address this assumption next for the D. W.Latham et al. (2010, in preparation) and CPM surveys, asthese are the primary techniques yielding new discoveries fromobservations of a subset of the overall sample. The speckle andHipparcos searches exhaustively covered the sample, and henceno extrapolation is necessary. We exclude the CCPS survey fromthis analysis because of the inherent bias in its sample selection,as it avoids known close binaries. The percentage of single starsamong the 344 primaries observed by D. W. Latham et al. (2010,in preparation) is 55% ± 4%, compared to 59% ± 7% for the109 stars not observed by this survey. These percentages agree

Relatively complete across q = Mcomp/M1 = 0.1 - 1.0 and log P (days) = 0 - 8

Detection techniques for companions to

solar-type primaries(Raghavan et al. 2010)

RV

LBI

AO

Speckle

CPM

logP(days)

q=M

comp/M

1loga(AU)

Detection Techniques for Companions to OB-type MS Primaries (M1 > 3 M☉)

Hurley 2006; Davis et al. 2010; Claeys et al. 2014). Moreover,MS binary distributions, such as the period and mass-ratiodistributions, may be significantly correlated with each other(Abt et al. 1990). Separately adjusting the input MS binarydistributions to the extremes may still not encompass the truenature of the binary population.

Companions to massive stars have been detected through avariety of methods, including spectroscopy (Sana et al. 2012),eclipses (Moe & Di Stefano 2015b), long-baseline interfero-metry (LBI; Rizzuto et al. 2013), adaptive optics (Shatsky& Tokovinin 2002), common proper motion (CPM; Abtet al. 1990), etc. Each observational technique is sensitive tocompanions across a certain interval of orbital periods P andmass ratios q≡Mcomp/M1 (see Figure 1). Despite significantadvances in the observational instruments and methods, theproperties of low-mass companions (q0.3) around early-type stars at intermediate orbital periods (P≈50–104 days)remain elusive. This region is especially interesting becauselow-mass X-ray binaries and millisecond pulsars that formin the galactic field (Kalogera & Webbink 1998; Kiel &Hurley 2006), as well as certain channels of Type Iasupernovae (Whelan & Iben 1973; Ruiter et al. 2011), mayevolve from extreme mass-ratio binaries at initially intermedi-ate orbital periods (Figure 1). Although we investigate allportions of the binary parameter space, we are especiallyconcerned with determining accurate companion statistics atintermediate orbital periods.

In order to use the properties of MS binaries to understandtheir formation and evolution, we must have a clear andcomplete description of young MS binaries. The primary goalof this study is to fit mathematical functions to the intrinsicdistributions and properties of a zero-age MS population ofbinary stars. We anchor our fits according to dozens of binary-star samples observed through a variety of techniques.However, each survey is affected by observational biases andincompleteness, and so we must first account for each survey’sparticular selection effects when measuring the intrinsicdistribution functions. In addition, because each survey issensitive to a narrow, specific interval of primary mass, orbitalperiod, and mass ratio, we must combine the statistics fromeach sample in a self-consistent, homogeneous manner. Onlythen can we understand binary-star populations as a whole andreliably measure the intrinsic interrelations between the binary-star properties.We organize the rest of this paper as follows. In Section 2,

we define the parameters that describe the statistics anddistributions of MS binary stars. We then review the variousobservational methods for detecting early-type binaries. Weanalyze spectroscopic binaries(Section 3), eclipsing binaries(EBs; Section 4), binaries discovered via LBI and sparseaperture masking (SAM; Section 5), binaries containing Cepheidprimaries that evolved from B-type MS stars (Section 6), andvisual binaries (VBs) identified through adaptive optics, luckyimaging, speckle interferometry, space-based observations,

Figure 1. Schematic diagram of the various observational techniques (regions enclosed with solid and dashed lines) used to identify companions to early-typeprimaries as a function of orbital period P and mass ratio q=Mcomp/M1. We compare the approximate parameter space of detection abilities for double-linedspectroscopic binaries (blue), eclipsing binaries (red), long-baseline interferometry (magenta), Cepheids (green), visual binaries (purple), X-ray emission (aqua), andCPM (orange). In this diagram, we show only observational techniques where the orbital period P (or orbital separation a) and mass ratio q can be estimated and thenature of the companion is reliably known to be a nondegenerate pre-MS or MS star. Assuming an average eccentricity á ñe ≈0.5, only binaries with log P(days)3.7 (left of dot-dashed line) will substantially interact via RLOF as the primary evolves toward the giant phase (see Section 11). Certain channels of low-mass X-ray binaries, millisecond pulsars, and Type Ia supernovae are expected to derive from early-type MS primaries with low-mass companions q≈0.1–0.3 atinitially moderate orbital periods P≈100–3000days (filled yellow region). We emphasize that only recent observations of eclipsing binaries (Moe & DiStefano 2015a, 2015b; red), companions identified via long-baseline interferometry (Rizzuto et al. 2013; Sana et al. 2014; magenta), and binaries with intermediate-mass Cepheid primaries (Evans et al. 2013, 2015; green) come close to probing this portion of the parameter space.

2

The Astrophysical Journal Supplement Series, 230:15 (55pp), 2017 June Moe & Di Stefano

Moe & Di Stefano (2017)Insensitive across small q & intermediate P – must correct for incompleteness!

Cannot directly measure multiplicity fractions Fbin, Ftrip, or Fquad.But can measure multiplicity frequency fmult = Fbin + 2Ftrip + 3Fquad.

Multiplicity Statistics: Diagnostics for Binary Star Formation (Abt+ 90; Kroupa 95a,b; Bate+ 95,02; Tokovinin 00,06; Tohline 02; Goodwin & Kroupa 05; Sana+ 12;

Kratter+ 06,10; Raghavan+ 10; Offner+ 12; Duchene & Krause 13; Tobin+ 16a,b; Moe & Di Stefano 17

Wide Companions:log P (days) = 5 - 9;a = 100 - 30,000 AU;Core Fragmentation

• fwide = 0.5, initially independent of M1

• f(q) initially consistent with randompairings drawn from IMF

• Subsequent dynamical ejections: systems with smaller M1 and q are preferentially disrupted by ZAMS







Intermediate-Period Companions:

log P (days) = 1 - 5;a = 0.1 - 100 AU;

Disk Fragmentation

• fmid = 0.4 (M1 = 1M☉) - 1.5 (30M☉)• f(q) correlated due to co-evolution / shared accretion in the disk

• M1 = 1M☉: uniform f(q) • M1 > 5M☉: weighted toward q = 0.2








log P (days) = 1 - 5;a = 0.1 - 100 AU;

Disk Fragmentation

Very Close Binarieslog P (days) < 1;

a < 0.1 AU;Dynamical Hardening in Triples during Pre-MS



• fclose = 0.02 (M1 = 1M☉) - 0.2 (30M☉)• Most have outer tertiaries• Uniform f(q) with excess twin fraction

Multiplicity Frequency

Solar-type MS (M1 = 1M☉): fmult = 0.50 ± 0.04;Fsingle = 60%, Fbin = 30%, Ftrip/quad = 10%

O-type MS (M1 = 28M☉): fmult = 2.1 ± 0.3;Fsingle < 5%, Fbin = 20%, Ftrip/quad = 75%

Corrected frequency of companions with q > 0.1

and log P (days) < 8 (a < 10,000 AU)

per primary near ZAMS

Adapted from Moe & Di Stefano 2017


For M1 = 1 M☉, frequency of wide companions (a > 100 AU)is 2 - 3 times larger during the early pre-MS phase (𝜏 < 3 Myr)

(Ghez et al. 1993; Duchene et al. 2007; Connelley et al. 2008; Tobin et al. 2016)

Overall multiplicity frequency: fmult = 0.5 → 0.8


Pre-MS (𝜏 < 3 Myr)


and log P (days) < 8 (a < 10,000 AU)



fmult ≈ 1.0 for all M1 < 1M☉ during pre-MS, but disruption of wide binaries with smaller binding energies (smaller M1 and q) reduces fmult by ZAMS

(Goodwin & Kroupa 2005; Marks & Kroupa 2011; Thies et al. 2015)




and log P (days) < 8 (a < 10,000 AU)



For M1 > 1M☉, disk fragmentation is progressively more likely with increasing primary mass (Kratter et al. 2006, 2011)




and log P (days) < 8 (a < 10,000 AU)


40

Fig. 36.— Top panel: Frequency flogP;q>0.3 (M1, P ) of companions with q > 0.3 per decade of orbital period. We display all ourmeasurements after correcting for incompleteness and selection effects, and we group the data into the same five primary mass / spectraltype intervals as displayed in Fig. 34. We fit analytic functions (dotted) to the observations. Solar-type binaries follow a log-normal perioddistribution with a peak of flogP;q>0.3 ≈ 0.08 near logP (days) = 5 (a ≈ 50AU). For early-type primaries, the companion frequenciesflogP;q>0.3 ≈ 0.1 - 0.2 are substantially larger at short (logP ! 1) and intermediate (2 ! logP ! 4) orbital periods. For mid-B and early-Bprimaries, the companion frequency peaks at log P ≈ 3.5 (a ≈ 10AU). For O-type stars, the orbital period distribution may be slightlybimodal with peaks at short (logP ! 1) and intermediate (log P ≈ 3.5) periods. Bottom panel: Frequency flogP;q>0.1 of companionswith q > 0.1 per decade of orbital period. In this case, our model for flogP;q>0.1 is completely described by the analytic functions that fitflogP;q>0.3 (top panel) and the mass-ratio distribution parameters Ftwin, γ largeq, and γsmallq (see Fig. 34). Although we do not directlyfit flogP;q>0.1, our analytic function matches the data reasonably well.

This effect is so important at long orbital periods that,although flogP;q>0.3 may be non-monotonic with respecttoM1 (see Eqn. 22 and right side of Fig. 36), flogP;q>0.1 ismonotonically increasing according to M1 for all orbitalperiods.Not all surveys we have examined in this study

are sensitive to binaries with q > 0.1, which is whywe have parameterized and measured the frequencyflogP;q>0.3 of companions with q > 0.3 (see Fig. 1and §2). Nonetheless, some samples are complete toq = 0.1 and so we can directly measure the companionfrequency flogP;q>0.1 down to q = 0.1. For solar-typeprimaries, the companion frequency is flogP;q>0.1 =(Nlargeq +Nsmallq)/Nprim, where Nprim = 404 and bothNlargeq and Nsmallq are given in Table 11 for each decadeof orbital period across 0 < logP < 8. Based on the Sana

et al. (2012) SB sample containing Nprim = 71 O-typeprimaries, we count Nlargeq = 17 companions withq > 0.3 and Nsmallq = 4 companions with q = 0.1 - 0.3across P = 2 - 20 days (see §3.5). This providesflogP;q>0.1 = (Nlargeq + Nsmallq)/Nprim = (17+4)/71 =0.30± 0.06. Similarly, for the combined SB sampleof Nprim = 81+109+83 = 273 B-type primaries, wemeasure flogP;q>0.1 = (Nlargeq + Nsmallq)Cevol/Nprim =(22+9)×1.2/273 = 0.14± 0.03 for logP = 0.8± 0.5.Based on observations of EBs with early-B primaries(see §4), we report in Moe & Di Stefano (2013) acorrected binary frequency of flogP;q>0.1 = 0.22± 0.05across P = 2 - 20 days. For the Nprim = 31 Cepheidsbrighter than V < 8.0 mag that were extensivelymonitored for radial velocity variations, Remage Evanset al. (2015) find Ncomp = 9 companions with

40





~2% of solar-type MS primaries have companions with q > 0.3 and P = 1 - 10 days.

Integral under dotted lines yields the MS multiplicity frequency fmult;q>0.3(M1).

Period distribution flogP;q>0.3(M1,P) from Moe & Di Stefano (2017)

40





40





Companions to solar-type MS stars: log-normal period distribution as found by Duquennoy & Mayor (1991) and Raghavan et al. (2010)


40





40





Very close binary fraction increases dramatically with M1(Abt et al. 1990; Sana et al. 2012; Chini et al. 2012; Kobulnicky et al. 2014)


40





40





Early-type MS stars also have a large companion frequency at intermediate P;Rizzuto+2013, LBI, early-B; Sana+2014, LBI, O-type; Evans+2015, SB2s, Cepheids

Disks around massive protostars are more prone to fragmentation (Kratter et al. 2006, 2011)


40





Frequency of wide companions with q > 0.3 relatively independent of M1,consistent with theories of core fragmentation

(Goodwin & Kroupa 2005; Offner et al. 2012; Thies et al. 2015)

40






Distribution fq(M1,P) of mass ratios q = Mcomp/M1

A single-component power-law model fq∝ q𝛾does NOT adequately describe the data. Let the data

be your guide!Need 3 parameters: 𝛾smallq(M1,P): power-law slope across q = 0.1 - 0.3𝛾largeq(M1,P): power-law slope across q = 0.3 - 1.0Ftwin(M1,P): excess fraction of twins with q ≈ 1.0

170 J. L. Halbwachs et al.: Statistical properties of F7–K binaries

0.0 0.5 1.0q

0.0

1.0

2.0

3.0

f q(q)

field binaries " ± σcluster binaries " ± σ

0.0 0.5 1.0q

0.0

1.0

2.0

3.0

f q(q)

all SBcorrected distribution " ± σ

0.0 0.5 1.0q

0.0

1.0

2.0

3.0

f q(q)

F7 - G " ± σK" ± σ

0.0 0.5 1.0q

0.0

1.0

2.0

3.0

f q(q)

P < 50 days " ± σP ≥ 50 days " ± σ

(a) (b)

(c) (d)

Fig. 7. Distribution of the mass ratios. As in the calculations, the distributions are drawn by joining the centers of the bins; the mass ratios rangeis divided in 10 equal bins in panel (a), and in 6 bins in the others. a) derived from the nearby SB and from the cluster SB with P < 10 years; theobserved distribution refers to all the SB, without any restriction about the ability to derive their orbital elements with CORAVEL; b) the F7–Gstars compared to the K-type stars; c) the binaries in open clusters compared to the binaries in the solar neighbourhood; d) the short–periodbinaries compared to the long–period binaries.

the number of peaks, the distribution is decreasing on the leftside, when small mass ratios are considered. The low frequencythat we found for small mass ratios even concerns compan-ions beyond the hydrogen–ignition limit: assuming a periodof 100 days, which is nearly the median of the sample, andthe median inclination i = 60 deg, a one solar mass binarywith q as small as 0.03 would have K1

p1 � e2 = 1.2 km s�1,

which corresponds roughly to the limit for deriving the orbit.Therefore, we find again the brown dwarf desert that was foundwith a larger sample of brown dwarf candidates (Halbwachset al. 2000). In that context, take note of the decrease of thedistribution of masses for the giant planets from 2 to 10 Jupitermasses, delimiting the low mass side of the desert of browndwarfs (Jorissen et al. 2001).

7.1.2. Does fq depend on the spectral type?

We implicitely assume that the formation process of stars isscale free, generating binaries obeying a certain mass ratio dis-tribution. However, the formation of the secondary componentscould depend on the mass of the primary, and the distribution ofM2 could be related toM1 – apart from the fact thatM2 M1.

Therefore, the distribution of mass ratios would depend on theprimary mass.

The sample was split in two parts, according to the primaryspectral types, and the intrinsic distributions of q were derivedagain. The results are in Fig. 7b. Apart from minor di↵erenceseasily explained by random variations, the distributions lookvery similar; each one is always within the 1 � interval ofthe other one. More surprisingly, the three peaks present in theoverall distribution are present again. Nevertheless, the inter-mediate peak cannot be attributed to white dwarf componentsfor the K-type SB, since the mass ratio should then be around 1.However, it is now rather flat, and it may appear simply bychance.

7.1.3. Is fq the same for cluster binaries as for fieldbinaries?

We had already checked on the raw data that the cluster SBobey the same distribution of mass ratios as the SB in the solarneighbourhood (Sect. 4.1). In order to compare the correcteddistributions, they were derived for the two samples and they

Spectroscopic FGK binaries

(Halbwachs et al. 2003)

𝛾smallq = 0.5𝛾largeq = –0.5Ftwin = 25%

Excess Twin Fraction Ftwin(M1,P)

(adapted from Moe & Di Stefano 2017)

Massive protostars have rapid contraction timescales and shorter disk lifetimes

Solar-type binaries: larger Ftwin due to pre-MS RLOF and/or shared accretion in longer-lived disks (Kroupa 1995; Tokovinin 2000; Halbwachs et al. 2003)

Mass-ratio distribution fq(P) for solar-type primaries M1 = 1M☉

companion properties to solar-type primaries may differin young star-forming environments (Duchêne et al. 2007;Connelley et al. 2008b; Kraus et al. 2011; Tobin et al. 2016), indense open clusters (Patience et al. 2002; Köhler et al. 2006;Geller & Mathieu 2012; King et al. 2012), or at extremely lowmetallicities (Abt 2008; Gao et al. 2014; Hettinger et al. 2015).In Section 10, we compare the corrected solar-type binarystatistics of the field MS population to those of younger MSand pre-MS populations in open clusters and stellar associa-tions. For now and for the purposes of binary populationsynthesis studies, we are mostly interested in the overallcompanion statistics of typical primaries with average ages.Most solar-type stars are near solar metallicity, in the galacticfield, and are several gigayears old. The volume-limited sampleof solar-type MS primaries in Raghavan et al. (2010) istherefore most representative of the majority of solar-type MSstars.

We display in Figure 28 the 168 confirmed companions fromRaghavan et al. (2010) with measured orbital periods0.0<log P (days)<8.0 and mass ratios 0.1<q<1.0. Weutilize the same methods as in Raghavan et al. (2010) toestimate the orbital periods P from projected separations andthe stellar masses from spectral types. Our Figure 28 is similarto Figures 11 and 17 in Raghavan et al. (2010). However, in thecases of triples and higher-order multiples, we always definethe period and mass ratio q≡Mcomp/M1 with respect to thesolar-type primary (see Section 2), which differs slightly fromthe definitions adopted in Raghavan et al. (2010).

For example, first consider a triple in an (Aa, Ab)–Bhierarchical configuration. An MAa=M1=1.0Me primaryand MAb=0.5Me companion are in a short-period orbitof P=100days. A long-period tertiary component withMB=0.4Me orbits the inner binary with a period ofP=105days. This system would contribute two data pointsin Figure 28: one with q=MAb/MAa=0.5 and log P=2.0,and one with q=MB/MAa=0.4 and log P=5.0. We do notdefine the mass ratio of the wide system to be q=MB/(MAa+ MAb), as done in Raghavan et al. (2010).

Next, consider a triple in an A–(Ba, Bb) hierarchicalconfiguration. A solar-type MA=M1=1.0Me primary is ina long-period P=105day orbit around a close, low-massbinary with MBa=0.5Me, MBa=0.4Me, and P=100days.In this situation, only the wide system with q=MBa/MA=0.5and log P=5.0 would contribute to our Figure 28. We do notconsider the low-mass inner binary with log P=2.0 andMBa/MBb=0.8, as done in Raghavan et al. (2010), becauseneither component Ba nor Bb is a solar-type star. Only ifcomponent Ba itself has an F6–K3 spectral type do we includethe close (Ba, Bb) pair in our sample. Nearly half of the twinswith q>0.95 in Figures 11 and 17 of Raghavan et al. (2010)are not solar-type twins. They instead contain late K or Mdwarf equal-mass binaries in a long-period orbit with a solar-type primary in an A–(Ba, Bb) hierarchical configuration. Our

Figure 28 therefore does not contain as many twin componentsas displayed in Figures 11 and 17 of Raghavan et al. (2010).

8.2. Corrections for Incompleteness

The Raghavan et al. (2010) sample is relatively completeexcept for two regions of the parameter space of P versus q.First, the survey is not sensitive to detecting companions withlog P≈5.9–6.7 and q≈0.1–0.2 (green region in ourFigure 28). As shown in Figure 11 of Raghavan et al.(2010), companions that occupy this portion in the parameterspace are missed by both adaptive optic and CPM techniques.Considering the density of systems in the immediatelysurrounding regions where the observations are relativelycomplete, we estimate that »4 additional systems occupy thisgap in the parameter space (four green systems in Figure 28).The second region of incompleteness occurs at q0.5 and

log P4.5 (blue region in our Figure 28). The opticalbrightness contrast between binary components is an evensteeper function of mass ratio for F–M type stars. Solar-typeSB2s with sufficiently luminous secondaries are observed onlyif q0.40–0.55 (brightness contrasts ΔV<5.0 mag, depend-ing on the orbital period). Spectroscopic binaries with lower-mass companions will generally appear as SB1s and thereforenot have mass ratios that can be readily measured. Of the fourspectroscopic binaries with log P<3.0 and q<0.4 shown inFigure 28, only one is an SB2 with P≈4days and a massratio q≈0.38 close to the detection limit. The other threesystems are SB1s in a hierarchical triple where the tertiary itselfhas an orbital solution. The total mass of the inner SB1 ismeasured dynamically, and so the mass of the companion inthe SB1 can be estimated. The few observed systems with3.0<log P<4.5 and q<0.4 are sufficiently nearby andhave favorable orientations for the companions to be resolved

Table 10Measurements of the Eccentricity Distribution Based on Early-type VBs withlog P (days)=3.6±0.5 from the Malkov et al. (2012) Visual Orbit Catalog

Primary Mass η

á ñM1 =3±1 Me 0.3±0.4

á ñM1 =11±4 Me 0.8±0.4

Figure 28. Companions to 454 solar-type stars from the Raghavan et al. (2010)survey as a function of P and q=Mcomp/M1. We display the 168 confirmedsystems (red plus signs) with measured mass ratios 0.1<q<1.0 and periods0<log P (days)<8. Two regions (blue and green lines) of this parameterspace are incomplete, either because the various observational techniques areinsensitive to these systems or because the systems in these regions aredetectable but have periods and/or mass ratios that cannot be readily measured(e.g., SB1s, radial velocity variables, and companions implied through proper-motion acceleration). We estimate 21 (blue diamonds) and 4 (green triangles)additional stellar MS companions located within these blue and green regions,respectively.

28


Data (Raghavan et al. 2010); corrections for selection biases & missing stellar companions ◇ and △ (Moe & Di Stefano 2017)

a < 100 AU:uniform fq with excess twins

(fragmentation & co-evolution in disk)

a > 100 AU:weighted toward

small q = 0.2 - 0.4(core fragmentation + dynamical ejections)

Mass-ratio distribution fq(P) for OB-type MS primaries M1 > 3M☉

orientations (Mazeh et al. 1992). However, SB1s with early-type MS primaries may not necessarily have low-mass A–Ktype stellar companions. Instead, many SB1s with O-type andB-type primaries may contain 1–3Me stellar remnants such aswhite dwarfs (WDs), neutron stars, or even black holes(Wolff 1978; Garmany et al. 1980). It is imperative to neverimplicitly assume that early-type SB1s contain two MScomponents. In the context of spectroscopic binaries withmassive primaries, only SB2s provide a definitive uncontami-nated census of unevolved companions (Moe & DiStefano 2015a). At the very least, SB1s can provide a self-consistency check and an upper limit to the frequency of low-mass stellar companions.

3.2. Corrections for Incompleteness

The ability to detect spectroscopic binaries depends not only onthe resolution of the spectrograph, the signal-to-noise ratio of thespectra, and the cadence of the observations but also on thephysical properties of the binary. Early-type stars, includingthose in binaries with P10days where tidal synchronizationis inefficient, are rotationally broadened by á ñvrot ≈100–200kms−1 (Abt et al. 2002; Levato & Grosso 2013). Theprimary’s orbital velocity semi-amplitude must therefore beK110kms−1 in order for the orbital reflex motion of the

primary to be detectable (Levato et al. 1987; Abt et al. 1990; Sanaet al. 2012; Kobulnicky et al. 2014). For SB2s, the atmosphericabsorption features of both the primary and secondary compo-nents need to be distinguishable. Due to blending of the broadabsorption features, early-type SB2s require an even higherthreshold of K140kms−1 to be observed. The primary’svelocity semi-amplitude K1 decreases toward wider separations a,smaller mass ratios q, and lower inclinations i. Lower-masscompanions at longer orbital periods are more likely to be missedin the spectroscopic binary surveys. Finally, highly eccentricbinaries spend only a small fraction of time near periastron whileexhibiting appreciable radial velocity variations. Depending on thecadence of the spectroscopic observations, eccentric binaries canbe either more or less complete compared to binaries in circularorbits.To measure the detection efficiencies and correct for

incompleteness, we utilize a Monte Carlo technique to generatea large population of binaries with different primary massesM1, mass ratios q, and orbital configurations P and e. For eachbinary, we select from a randomly generated set of orientations,i.e., an inclination i distribution such that cos i=[0, 1] isuniform and a distribution of periastron angles ω=[0o, 360o]that is also uniform. The velocity semi-amplitude K1 criterionused in previous studies does not adequately describe thedetection efficiencies of eccentric binaries. For each binary, weinstead synthesize radial velocity measurements v r1, and v r2, at20 random epochs, which is the median cadence of thespectroscopic binary surveys (Levato et al. 1987; Abtet al. 1990; Sana et al. 2012; Kobulnicky et al. 2014). Forsimplicity, we assume that the systemic velocity of the binary iszero. In an individual spectrum of an early-type star, the radialvelocities can be centroided to an accuracy of≈2–3kms−1.However, atmospheric variations limit the true sensitivityacross multiple epochs to dv r1, ≈3–5kms−1 (Levatoet al. 1987; Abt et al. 1990; Sana et al. 2012; Kobulnickyet al. 2014). In order for a simulated binary to have an orbitalsolution that can be fitted, and therefore an eccentricity andmass ratio that can be measured, we require a minimumnumber of radial velocity measurements of the primary v r1, tosignificantly differ from the systemic velocity. We imposethat at least 5 of the 20 measurements satisfy ∣ ∣v r1, (3–5)dv r1, ≈15kms−1 in order to provide a precise andunique orbital solution for the primary.The rotationally broadened spectral features of both the primary

and secondary must also be distinguishable to be cataloged as anSB2. At the very least, the primary’s absorption features must notonly shift during the orbit but also have velocity profiles thatvisibly change as a result of the moving absorption lines from theorbiting secondary (De Becker et al. 2006; Sana et al. 2012). Forq<0.8, we require at least 3 of the 20 measurements to satisfy

-∣ ∣v vr r1, 2, á ñv 2rot ≈75kms−1. This velocity threshold isappropriate for both O-type and B-type primaries because themean rotational velocitiesá ñvrot ≈150kms−1 do not significantlyvary between these two spectral types (Abt et al. 2002; Levato &Grosso 2013). For twin binaries with q=1.0, the absorption linesof both the primary and secondary are of comparable depths, andso it is more difficult to distinguish the two components if theirabsorption features are significantly blended (Sana et al. 2011).For these twin systems, we require at least 3 of the 20measurements to satisfy -∣ ∣v vr r1, 2, >á ñvrot =150kms−1. Welinearly interpolate our velocity difference thresholds between75kms−1 at q=0.8 and 150kms−1 at q=1.0.

Figure 4. Similar to Figure 3, but for the mass ratios q=M2/M1 as a functionof orbital period P. Early-type SB2s with MS components can reveal onlycompanions with q>0.25 (above dotted line). Assuming M1=10 Me,e= e0.5 max , and the median cadence and sensitivity of the spectroscopicsurveys, we show completeness levels of 80%, 50%, and 20% (dashed). Short-period SB2s with P<20 days span the entire observable mass-ratio intervalq≈0.25–1.0. Alternatively, long-period systems with P=100–500daysinclude only SB2s with q≈0.3–0.4, even though the spectroscopic surveysare substantially incomplete in this corner of the parameter space.

Table 3Binary Statistics Based on 23 Early-type SB2s with log P (days)=1.3±0.3

Contained in the SB9 Catalog

Sample Primary Mass Statistic

Spectral Types O9–B3 á ñM1 =14±4 Me η=0.9±0.4g qlarge =−0.6±0.9

Spectral Types B5–B9.5 á ñM1 =4.5±1.5 Me η=−0.3±0.2g qlarge =−1.0±0.8

7


P < 20 days: 𝛾largeq = –0.4;

Ftwin = 10 - 20%

P = 20 - 500 days:𝛾largeq = –1.6;

Ftwin < 5%

(Moe & Di Stefano 2017)

SB2sonly

Power-law component 𝛾largeq(M1,P) of mass-ratio distribution fq

Solar-type MS binaries (Raghavan et al. 2010);SB2 companions to O / early-B MS primaries (Abt+90; Sana+12; Kobulnicky+14)

Moe & Di Stefano 17


Eclipsing binaries with early-B primaries (Moe+ 2013; 2015)

Moe & Di Stefano 17


Radial velocity companions to Cepheid primaries(Evans+ 2015)

Moe & Di Stefano 17


Long-baseline interferometry of early-B (Rizzuto+ 2013) and O-type (Sana+ 2014) MS primaries

Moe & Di Stefano 17


Visual companions to early-type MS primaries,including CPM (Abt+ 1990), AO (Duchene+ 2011; Sana+ 2014),

lucky imaging (Peter+ 2012), and HST imaging (Aldoretta+ 2015)

Moe & Di Stefano 17


Moe & Di Stefano 17

Very close binaries (a < 0.1 AU): uniform fq with excess twin fraction.Wide binaries (a > 100 AU): initially consistent with random pairings from IMF.

Transition occurs at shorter separations for more massive binaries.


The mass-ratio distribution fq of binaries that will interact (a < 10 AU; left of dotted line) depends critically on M1 and P

Moe & Di Stefano 17

function takes advantage of the information provided by allbinary orbits and provide a more robust determination of thecircularization period for the typical binary orbit.

7. THE CIRCULARIZATION PERIODS OF SEVENADDITIONAL BINARY POPULATIONS

Below we briefly discuss and present the period-eccentricitydistributions and determine circularization periods for seven ad-ditional binary populations spanning ages from !3 Myr (PMSbinaries) to !10 Gyr (Galactic halo binaries). Figures 8a–8hshow the orbital data of each individual population with the best-fit circularization function overplotted and the circularization

period with error marked. All results are listed in Table 3; here webriefly discuss each population in turn.PMS binary population (Fig. 8a).—The sample of 37 low-

mass PMS binaries show the characteristic period overlap be-tween eccentric and circular orbits. The orbital parameters aretaken from Melo et al. (2001), and references to individualbinaries can be found in their paper. The PMS sample is notstrictly coeval, but covers an age range from !1–10 Myr (Meloet al. 2001). We determine a circularization period of 7:1þ1:2

#1:2 daysfor the PMS binary sample. This value should be compared tothe 7.56 day cutoff period determined by Melo et al. Note thatMelo et al. chose to disregard the circular orbit of binary

Fig. 8.—Period-eccentricity distributions for eight late-type binary populations: (a) PMS, (b) Pleiades, (c) M35, (d ) Hyades/Praesepe, (e) M67, ( f ) NGC 188, (g)field, and (h) halo. The best-fit circularization function is plotted over each distribution. A solid horizontal line marks e(P) ¼ 0:00. The CP and its uncertainty aremarked on the period axis by a vertical black line and a gray bar, respectively. The circularization period and the age of the binary population are printed in each plot.

MEIBOM & MATHIEU978 Vol. 620

Tidal circularization period increases from Pcirc = 6 days (pre-MS) to Pcirc = 15 days (halo)

Beyond P > 100 days,fe∝ eη with η = 0.4

Eccentricity distribution for solar-type binaries (Meibom & Mathieu 2005)

Eccentricity distribution for early-type binaries

Pcirc = 2 days (tides less efficientin early-type stars)

á ñM1 ≈28Me; 24SB2s). We note that the Kobulnicky et al.(2014) survey also includes O-type stars, but we consider onlytheir 83 systems with early B primaries in our current analysis.

We list the multiplicity statistics based on the four samples inTable 2. In Figures 3 and4, we display the eccentricities e andmass ratios q, respectively, of the 44 SB2s as a function oforbital period. Sana et al. (2012) and Kobulnicky et al. (2014)identified additional SB2s with q>0.55 and P>500days,which confirms that spectroscopic observations can revealmoderate mass-ratio binaries at intermediate orbital periods.However, spectroscopic binaries become increasingly lesscomplete and biased toward moderate q at longer P (seeSection 3.2 and Figure 4), so we limit our sample selection toSB2s with P< 500days when discussing these four surveys.

We also examine the 23 detached SB2s with primaryspectral types O and B, luminosity classes III–V, orbitalperiods P=8–40days, and measured mass ratios q=K1/K2and eccentricities e from the Ninth Catalog of SpectroscopicBinaries (SB9; Pourbaix et al. 2004). This sample includes 10systems with O9–B3 primaries (á ñM1 ≈14Me) and 13 systemswith B5–B9.5 primaries (á ñM1 ≈4.5Me). We report themultiplicity statistics based on these 23 SB2s in Table 3.The spectroscopic binaries contained in the SB9 catalogare compiled from a variety of surveys and samples, and sothe cadence and sensitivity of the spectroscopic observationsare not as homogeneous. We therefore consider only the SB2swith P<40days, which are relatively complete regardless ofthe instruments utilized. We cannot infer the intrinsic frequencyof companions per primary based on the SB9 catalog. We cannonetheless utilize the sample of 23 early-type SB2s to measurethe eccentricity and mass-ratio distributions.

For SB2s, the secondary must be comparable in luminosityto the primary in order for both components to be visible in the

combined spectrum. Because MS stars follow a steep mass–luminosity relation, SB2s with early-type MS primaries canreveal only moderate mass ratios q>0.25 (Figures 1 and 4).For single-lined spectroscopic binaries (SB1s) with low-luminosity companions, a lower limit to the mass ratio canbe estimated from the observed reflex motion of the primary. Astatistical mass-ratio distribution can be recovered for SB1s byassuming that the intrinsic binary population has random

Table 2Statistics Based on Four Surveys Containing Early-type Spectroscopic Binaries with log P (days)=0.3–2.7

Survey Primary Mass Period Interval Statistic

All Four á ñM1 =16±8 Me log P (days)=0.75±0.25 η=−0.3±0.2” ” log P (days)=1.85±0.85 η=0.6±0.3” ” log P (days)=0.8±0.5 g qlarge =−0.3±0.3

” ” log P (days)=2.0±0.7 g qlarge =−1.6±0.5

” ” ” �twin<0.05

Levato et al. (1987) and á ñM1 =6±2 Me log P (days)=0.8±0.5 �twin=0.17±0.09Abt et al. (1990) ” ” ”

Kobulnicky et al. (2014) and á ñM1 =20±7 Me ” �twin=0.08±0.04Sana et al. (2012) ” ” ”

Levato et al. (1987), á ñM1 =8±3 Me log P(days)=0.8±0.5 g qsmall =−0.5±0.8

Abt et al. (1990), and ” ” ”

Kobulnicky et al. (2014) ” ” ”

Sana et al. (2012) á ñM1 =28±8 Me ” g qsmall =0.6±0.8

Levato et al. (1987) á ñM1 =5±2 Me log P (days)=0.8±0.5 >f P qlog ; 0.3=0.07±0.04

Abt et al. (1990) á ñM1 =7±2 Me ” >f P qlog ; 0.3=0.10±0.04

Kobulnicky et al. (2014) á ñM1 =12±3 Me ” >f P qlog ; 0.3=0.12±0.05

Sana et al. (2012) á ñM1 =28±8 Me ” >f P qlog ; 0.3=0.24±0.06

Abt et al. (1990) and á ñM1 =9±3 Me log P (days)=1.8±0.5 >f P qlog ; 0.3=0.08±0.05

Kobulnicky et al. (2014) ” ” ”

Sana et al. (2012) á ñM1 =28±8 Me log P (days)=2.0±0.7 >f P qlog ; 0.3=0.12±0.06

Figure 3. Eccentricities e of 44 SB2s as a function of orbital period P from fourspectroscopic surveys: Sana et al. (2012; magenta plus signs), Kobulnicky et al.(2014; blue crosses), Abt et al. (1990; green diamonds), and Levato et al.(1987; red squares). We display the maximum expected eccentricity emax (P)according to Equation (3) (black line). Assuming M1=10Me, q=0.4,random orientations, and the median cadence and sensitivity of the spectro-scopic surveys, we show completeness levels of 80%, 50%, and 20% (dashed).Note that many SB2s with P<10 days are circularized (e=0; dotted), whileall early-type SB2s with P>10days are in eccentric orbits.

6


Moe & Di Stefano 17

emax(P): maximum eccentricity without filling Roche-lobe at

periastron

Let the data be your guide!

Eccentricity distribution fe (M1,P)

Tidal circularization dominates at P < 20 daysFor P > 20 days, early-type binaries are consistent with a thermal distribution (η = 1; Ambartsumian 1937), indicating dynamical interactions play a role in their formation

37

For mid-B, early-B, and O-type stars, we fit:

γsmallq (M1 > 6.0M⊙, P ) =

0.1 for 0.2 ≤ logP < 1.0,

0.1− 0.15(logP − 1) for 1.0 ≤ logP < 3.0,

−0.2− 0.50(logP − 3) for 3.0 ≤ logP < 5.6,

−1.5 for 5.6 ≤ logP < 8.0. (15)

As done previously, we interpolate between Eqn. 13 andEqn. 14 for primary stars with masses M1 = 1.2 - 3.5M⊙.Similarly, for M1 = 3.5 - 6.0M⊙, we interpolate betweenEqn. 14 and Eqn. 15 with respect to M1.We compare our fit to the data for γsmallq in the

bottom panel of Fig. 34. Our fit passes through all themeasurements within their respective 1σ uncertainties.The uncertainty in the fit is therefore dominated bythe uncertainty in the observations. The 1σ errors inγsmallq depend primarily on P . They increase fromδγsmallq ≈ 0.4 at logP ≈ 1 to δγsmallq ≈ 0.6 atlogP ≈ 3, and then they decrease to δγsmallq ≈ 0.3 forlogP ! 6. For all primary masses M1, we model the 1σuncertainties as:

δγsmallq (M1, P ) =

0.4 for 0.2 ≤ logP < 1.0,

0.4 + 0.1(logP − 1) for 1.0 ≤ logP < 3.0,

0.6− 0.1(logP − 3) for 3.0 ≤ logP < 6.0,

0.3 for 6.0 ≤ logP < 8.0. (16)

This relation accounts for the larger uncertainties inγsmallq at intermediate orbital periods due to the gapsin the observations.

9.2. Eccentricity Distributions

In Fig. 35, we display the power-law slope η ofthe eccentricity distribution as a function of logP andcolored according to spectral type based on all ourmeasurements in Tables 2 - 12. We divide the data intotwo spectral type intervals: late-type (M1 = 0.8 - 5M⊙)and early-type (M1 > 5M⊙). Both late-type (Meibom& Mathieu 2005; Raghavan et al. 2010) and early-type(Sana et al. 2012) populations have similar zero-age MScircularization periods P ≈ 2 - 6 days (logP ≈ 0.5). Inaddition, for both late-type and early-type binaries, theeccentricity distribution becomes weighted toward largervalues with increasing orbital period P . However, thepower-law slope is ∆η ≈ 0.5 larger for early-type binariescompared to late-type binaries across all orbital periods.For close binaries with logP " 1.2, the differences in

the eccentricity distribution between the two spectraltypes can be explained by tidal evolution. Tidal dampingvia convection is more efficient in cool, late-type starsthan is radiative damping in hot, early-type stars (Zahn1975, 1977; Hut 1981). Moreover, late-type MS binarieslive orders-of-magnitude longer and have had more timeto tidally evolve toward smaller eccentricities.For binaries with intermediate periods 1.2 " logP " 5,

however, the differences cannot be explained solely bytidal effects. In §8.6, we showed that tidal evolution

Fig. 35.— The power-law slope η of the eccentricity distributionpe ∝ eη across eccentricities 0.0 < e < emax such that thecomponents are not Roche-lobe filling. We compare our correctedmeasurements of η as a function of orbital period P and coloredaccording to primary spectral type: late (M1 = 0.8 - 5M⊙;red) and early (M1 > 5M⊙; blue). We fit the data (dotted)for the late-type and early-type populations. Although bothlate-type and early-type binaries become weighted toward largereccentricities with increasing period, early-type binaries quicklyasymptote toward a thermal eccentricity distribution (η = 1)beyond logP (days) ! 1. Meanwhile, solar-type binaries asymptotetoward η ≈ 0.5, about halfway between the uniform (η = 0) andthermal (η = 1) distributions (dashed lines).

is negligible for MS solar-type binaries with periodslogP = 1.2 - 2.4 slightly beyond the tidal circularizationperiod. For late-type binaries, even those with orbitalperiods 3 < logP < 5 where tidal effects are even lesssignificant, the power-law slope η ≈ 0.3 - 0.6 is discrepantwith a thermal eccentricity distribution (see Fig. 35). Atwider separations ⟨a⟩ ≈ 120 AU (logP ≈ 5.6), Tokovinin& Kiyaeva (2015) demonstrate that solar-type binarieshave an intrinsic eccentricity probability distributiondescribed by pe = 1.2e+0.4. Fitting our power-law modelto their data (Fig. 7 in Tokovinin & Kiyaeva 2015),we measure η = 0.5± 0.3. We confirm the conclusionof Tokovinin & Kiyaeva (2015) that the eccentricitydistribution of wide solar-type binaries is flatter thana thermal distribution. Meanwhile, for early-typebinaries, the initial MS eccentricity distribution quicklyasymptotes toward a thermal distribution (η = 1)beyond logP ! 1 (Fig. 35). Early-type binaries withintermediate orbital periods are born on the zero-ageMS with systematically larger eccentricities than theirsolar-type counterparts.We adopt an analytic function for η that approaches

circular orbits for short-period binaries and thenasymptotes toward a fixed η at longer periods. Forlate-type binaries, we use:

η (0.8M⊙ < M1 < 3M⊙, 0.5 < logP < 6.0) =

0.6− 0.7

logP − 0.5. (17)

Meanwhile, for early-type binaries, we fit:

η (M1 > 7M⊙, 0.5 < logP < 5.0) =

0.9− 0.2

logP − 0.5. (18)

fe∝ eη

across domain0 < e < emax(P)

Moe & Di Stefano 17

(5 + 11)/418/1.0=0.038±0.010 companions with q>0.1per decade of orbital period. The remaining 14 of our selectedSB1s reside across P=200–1000days (Δ logP=0.7), wherethe observations are slightly incomplete toward systems withsmall mass ratios q=0.1–0.4. We estimate a small overallcorrection factor »1.2 to account for the incompleteness atthese intermediate separations, and so the companion frequencyis >f P qlog ; 0.1=14×1.2/418/0.7=0.057±0.015. We display these three data points based on SBcompanions to solar-type MS stars in M35 as red data points inFigure 41.

Patience et al. (2002) utilized ground-based speckle imagingand direct imaging with HST to resolve companions to B–K MSstars in both the αPersei (90Myr) and Praesepe (660Myr) openclusters. We select their 79 systems with K=8.5–10.5mag(F5–K5 primary spectral types) in the younger αPer cluster. Forthese primaries, Patience et al. (2002) identified 12 companionswith q>0.1 (ΔK5.5 mag) across projected separationsρ=0 3–5″, i.e., a≈50–800 au (logP=5.0–6.8) given thedistance d≈180 pc to αPer. Their survey was relativelycomplete down to q=0.1 (ΔK≈5.5 mag) across this separa-tion range, and so we measure >f P qlog ; 0.1=13/79/(6.8− 5.0)=0.091±0.025 companions with q>0.1 perdecade of orbital period across logP (days)=5.0–6.8 (orangedata point in Figure 41).

As can be seen in Figure 41, the frequency of companions tosolar-type MS stars in young open clusters is consistent with

that measured in the field population across both short(logP<3) and long (logP=5–7) orbital periods. Othersurveys have also concluded that the statistics of solar-typebinaries in open clusters are indistinguishable from those in thefield for a broad range of cluster densities and ages τ≈3Myr–7 Gyr (Bouvier et al. 1997; Köhler et al. 2006; Krauset al. 2011; Geller & Mathieu 2012; King et al. 2012). Thisdemonstrates that the formation of solar-type binaries isrelatively universal and that there is negligible evolution ofthe solar-type MS binary statistics for ages τ3Myr.We next turn our attention to wide companions to pre-MS

protostars with ages τ3Myr. Unlike MS binaries, where themeasured brightness contrasts map robustly to mass ratios,accretion luminosity in pre-MS stars can dominate over thephotospheric flux. Accounting for this effect, Connelley et al.(2008a) estimate that a near-infrared brightness contrastΔL=4mag roughly corresponds to q≈0.1 for coeval pre-MS binaries on the Hayashi track.Duchêne et al. (2007) employed near-IR adaptive optics to

search for wide companions to 45 ClassI and flat-spectrumprotostars embedded in four different molecular clouds. Theyidentified 15 physically associated companions with separationsρ=0 2–10 0 and brightness contrastsΔL<4mag (q0.1).The Duchêne et al. (2007) sample is complete to ΔL<4.0magcompanions across this separation range, which corresponds toa≈40–2000 au (logP=4.9–7.4) given the average distanced≈200 pc to the four molecular clouds. Of our 15 selectedbinaries, 6 have ρ=0 2–1 2 (logP=4.9–6.1) and theremaining 9 have ρ=1 2–10″ (logP=6.1–7.4). We measure

>f P qlog ; 0.1=6/45/(6.1− 4.9)=0.11±0.04 across logP(days)=4.9–6.1 and >f P qlog ; 0.1=9/45/(7.4− 6.1)=0.15±0.05 across logP (days)=6.1–7.4 (blue data points inFigure 41).Connelley et al. (2008a) observed a much larger sample of

189 Class I young stellar objects (YSOs) in the near-infrared.They identified a total of 65 companions with separationsρ=0 3–10″ and brightness contrasts ΔL<4mag (q0.1).We note that 13 of their YSOs have two or even three resolvedcompanions that reside in the narrow interval ρ=0 3–10″,contributing 31 of the 65 total companions in our statistic. Asignificant fraction of these triples and quadruples in which thecompanions all have similar separations are most likelygravitationally unstable in their current configurations. If so,either they will dynamically evolve into stable hierarchicalconfigurations, or one of the components will get ejected (seemore below). The Connelley et al. (2008a) survey is completeto ΔL=4mag (q≈0.1) across our selected intervalρ=0 3–10″, which corresponds to a=150–5000 au(logP=5.7–8.0) given the average distance d≈500pc tothe YSOs. We divide our 65 companions across three intervals:18 with ρ=0 3–1 0 ( >f P qlog ; 0.1=0.12±0.03 acrosslogP=5.7–6.5), 25 companions with ρ=1 0–3 0( >f P qlog ; 0.1=0.18±0.04 across logP=6.5–7.2), and 22companions with ρ=3 0–10 0 ( >f P qlog ; 0.1=0.15±0.03across logP=7.2–8.0). We display these three data points inmagenta in Figure 41.Finally, we analyze the spectroscopic binary survey of pre-

MS T Tauri stars conducted by Melo (2003), who updated andextended the sample of Mathieu (1992, 1994). Melo (2003)identified four SBs with P=2–200 days (Δ logP=2.0)within their sample of 65 TTauri stars. According to theirFigure 2, the Melo (2003) survey is relatively complete toward

Figure 41. Frequency >f P qlog ; 0.1 of companions with q>0.1 per decade oforbital period for various populations of primary stars after correcting forselection effects and incompleteness. In black, we compare the distributionbased on the volume-limited sample of solar-type MS stars in the field (solid) toour analytic fits for the solar-type MS (dotted) and O-type MS (dashed)populations. We also display the binary properties of solar-type stars in youngopen clusters (red and orange) and of pre-MS solar-type stars (blue, magenta,and cyan). For wide binaries (log P=6–8; a=200–5000 au) that form viafragmentation of cores/filaments, the multiplicity statistics of solar-type pre-MS stars is consistent with that measured for massive MS stars. Subsequentdynamical processing that preferentially ejects q=0.1–0.3 companionsreduces the overall companion frequency of solar-type binaries at theseseparations. For closer binaries (log P<6; a<200 au) that form via diskfragmentation, the solar-type MS and pre-MS companion frequencies areconsistent with each other, both of which are significantly smaller than themeasured O-type MS companion frequencies. A primary-mass-dependentprocess, e.g., more massive protostars are more prone to disk fragmentation, isrequired to explain why massive stars have larger companion frequencies atshort and intermediate separations.

47


O-type MS

Period distribution flogP;q>0.1(M1, P, 𝜏) from Moe & Di Stefano (2017)

Frequency of wide companions (a > 100 AU) to M1 = 1M☉ pre-MS primaries is 2 - 3 times larger than that measured for M1 = 1M☉ MS stars (Duchene+07, Connelley+08)

and consistent with that measured for O-type MS primaries

G-type MS

Frequency ofcompanions with

q > 0.1 per decade of period

(5 + 11)/418/1.0=0.038±0.010 companions with q>0.1per decade of orbital period. The remaining 14 of our selectedSB1s reside across P=200–1000days (Δ logP=0.7), wherethe observations are slightly incomplete toward systems withsmall mass ratios q=0.1–0.4. We estimate a small overallcorrection factor »1.2 to account for the incompleteness atthese intermediate separations, and so the companion frequencyis >f P qlog ; 0.1=14×1.2/418/0.7=0.057±0.015. We display these three data points based on SBcompanions to solar-type MS stars in M35 as red data points inFigure 41.

Patience et al. (2002) utilized ground-based speckle imagingand direct imaging with HST to resolve companions to B–K MSstars in both the αPersei (90Myr) and Praesepe (660Myr) openclusters. We select their 79 systems with K=8.5–10.5mag(F5–K5 primary spectral types) in the younger αPer cluster. Forthese primaries, Patience et al. (2002) identified 12 companionswith q>0.1 (ΔK5.5 mag) across projected separationsρ=0 3–5″, i.e., a≈50–800 au (logP=5.0–6.8) given thedistance d≈180 pc to αPer. Their survey was relativelycomplete down to q=0.1 (ΔK≈5.5 mag) across this separa-tion range, and so we measure >f P qlog ; 0.1=13/79/(6.8− 5.0)=0.091±0.025 companions with q>0.1 perdecade of orbital period across logP (days)=5.0–6.8 (orangedata point in Figure 41).

As can be seen in Figure 41, the frequency of companions tosolar-type MS stars in young open clusters is consistent with

that measured in the field population across both short(logP<3) and long (logP=5–7) orbital periods. Othersurveys have also concluded that the statistics of solar-typebinaries in open clusters are indistinguishable from those in thefield for a broad range of cluster densities and ages τ≈3Myr–7 Gyr (Bouvier et al. 1997; Köhler et al. 2006; Krauset al. 2011; Geller & Mathieu 2012; King et al. 2012). Thisdemonstrates that the formation of solar-type binaries isrelatively universal and that there is negligible evolution ofthe solar-type MS binary statistics for ages τ3Myr.We next turn our attention to wide companions to pre-MS

protostars with ages τ3Myr. Unlike MS binaries, where themeasured brightness contrasts map robustly to mass ratios,accretion luminosity in pre-MS stars can dominate over thephotospheric flux. Accounting for this effect, Connelley et al.(2008a) estimate that a near-infrared brightness contrastΔL=4mag roughly corresponds to q≈0.1 for coeval pre-MS binaries on the Hayashi track.Duchêne et al. (2007) employed near-IR adaptive optics to

search for wide companions to 45 ClassI and flat-spectrumprotostars embedded in four different molecular clouds. Theyidentified 15 physically associated companions with separationsρ=0 2–10 0 and brightness contrastsΔL<4mag (q0.1).The Duchêne et al. (2007) sample is complete to ΔL<4.0magcompanions across this separation range, which corresponds toa≈40–2000 au (logP=4.9–7.4) given the average distanced≈200 pc to the four molecular clouds. Of our 15 selectedbinaries, 6 have ρ=0 2–1 2 (logP=4.9–6.1) and theremaining 9 have ρ=1 2–10″ (logP=6.1–7.4). We measure

>f P qlog ; 0.1=6/45/(6.1− 4.9)=0.11±0.04 across logP(days)=4.9–6.1 and >f P qlog ; 0.1=9/45/(7.4− 6.1)=0.15±0.05 across logP (days)=6.1–7.4 (blue data points inFigure 41).Connelley et al. (2008a) observed a much larger sample of

189 Class I young stellar objects (YSOs) in the near-infrared.They identified a total of 65 companions with separationsρ=0 3–10″ and brightness contrasts ΔL<4mag (q0.1).We note that 13 of their YSOs have two or even three resolvedcompanions that reside in the narrow interval ρ=0 3–10″,contributing 31 of the 65 total companions in our statistic. Asignificant fraction of these triples and quadruples in which thecompanions all have similar separations are most likelygravitationally unstable in their current configurations. If so,either they will dynamically evolve into stable hierarchicalconfigurations, or one of the components will get ejected (seemore below). The Connelley et al. (2008a) survey is completeto ΔL=4mag (q≈0.1) across our selected intervalρ=0 3–10″, which corresponds to a=150–5000 au(logP=5.7–8.0) given the average distance d≈500pc tothe YSOs. We divide our 65 companions across three intervals:18 with ρ=0 3–1 0 ( >f P qlog ; 0.1=0.12±0.03 acrosslogP=5.7–6.5), 25 companions with ρ=1 0–3 0( >f P qlog ; 0.1=0.18±0.04 across logP=6.5–7.2), and 22companions with ρ=3 0–10 0 ( >f P qlog ; 0.1=0.15±0.03across logP=7.2–8.0). We display these three data points inmagenta in Figure 41.Finally, we analyze the spectroscopic binary survey of pre-

MS T Tauri stars conducted by Melo (2003), who updated andextended the sample of Mathieu (1992, 1994). Melo (2003)identified four SBs with P=2–200 days (Δ logP=2.0)within their sample of 65 TTauri stars. According to theirFigure 2, the Melo (2003) survey is relatively complete toward

Figure 41. Frequency >f P qlog ; 0.1 of companions with q>0.1 per decade oforbital period for various populations of primary stars after correcting forselection effects and incompleteness. In black, we compare the distributionbased on the volume-limited sample of solar-type MS stars in the field (solid) toour analytic fits for the solar-type MS (dotted) and O-type MS (dashed)populations. We also display the binary properties of solar-type stars in youngopen clusters (red and orange) and of pre-MS solar-type stars (blue, magenta,and cyan). For wide binaries (log P=6–8; a=200–5000 au) that form viafragmentation of cores/filaments, the multiplicity statistics of solar-type pre-MS stars is consistent with that measured for massive MS stars. Subsequentdynamical processing that preferentially ejects q=0.1–0.3 companionsreduces the overall companion frequency of solar-type binaries at theseseparations. For closer binaries (log P<6; a<200 au) that form via diskfragmentation, the solar-type MS and pre-MS companion frequencies areconsistent with each other, both of which are significantly smaller than themeasured O-type MS companion frequencies. A primary-mass-dependentprocess, e.g., more massive protostars are more prone to disk fragmentation, isrequired to explain why massive stars have larger companion frequencies atshort and intermediate separations.

47


O-type MS

G-type MS

For pre-MS (Mathieu 94; Melo 03) and solar-type MS primaries in open clusters (Patience+03; Geller+12; Leiner+15), the companion

frequency across a < 100 AU matches that for field solar-type MS stars, which is substantially smaller than that measured for O-type MS stars.

Period distribution flogP;q>0.1(M1, P, 𝜏) from Moe & Di Stefano (2017)

Frequency ofcompanions with

q > 0.1 per decade of period

Very close binaries derive from dynamical interactions in triples

45

pre-MS phase is a third option to explain the correlatedcomponent masses and larger excess twin fraction.In any case, we emphasize the transition in the

mass-ratio distribution between short (a ! 0.3 AU)and intermediate (a ≈ 50 AU) separations is morepronounced for more massive systems (Fig. 34).For early-type binaries, the excess twin fractionvanishes beyond P > 20 days and the power-lawcomponent dramatically decreases from γlargeq ≈ −0.5to γlargeq ≈ −2.0. Meanwhile, for solar-type binaries, theexcess twin fraction decreases slightly from Ftwin ≈ 0.3 toFtwin = 0.1 and the power-law component γlargeq ≈ −0.5remains relatively constant. This intrinsic variation withrespect to primary mass may be due to the scalingof the fragment mass ratio qfrag discussed above, butmay also be due to the longer formation timescalesassociated with lower mass primaries. For example,the average primordial disk lifetimes τdisk = 3 Myrof solar-type primaries (Mamajek 2009) is an orderof magnitude longer than the disk photoevaporationtimescales τdisk ! 0.3 Myr measured in more massiveHerbig Be stars (Alonso-Albi et al. 2009). The longerdisk lifetimes of solar-type systems may allow a largerfraction of companions to accrete relatively more massfrom the disk, possibly toward q ≈ 1. Similarly,the pre-MS contraction timescales is significantly longerfor solar-type binaries, and so short-period solar-typebinaries are more likely to exchange material throughRoche-lobe overflow while on the pre-MS. This mayexplain the correlation between the excess twin fractionand primary mass at short orbital periods P < 10 days.In summary, the processes of disk fragmentation,accretion in the disk, and pre-MS mass transfer mayall contribute to the larger excess twin fraction andhigher degree of correlation between component massesobserved in solar-type binaries compared to early-typebinaries.Although the correlation between binary component

masses demonstrate they coevolved as they migratedtoward shorter separations, they do not reveal preciselyhow the companions migrated. It is possible thatcompanions naturally undergo orbital decay towardsmaller separations due to hydrodynamical forcesin the disk. It is also possible that the innerbinary requires an outer tertiary to evolve towardshorter periods. After considering selection effectsand accounting for incompleteness, Tokovinin et al.(2006) show that ≈(70 - 90)% of solar-type binarieswith periods P ≈ 2 - 6 days have outer tertiaries withq = M3/M1 " 0.2. Meanwhile, only ≈30% of binarieswith P ≈ 10 - 30 days have such tertiary components.Tokovinin et al. (2006) argue that close binaries formpredominantly through Kozai cycles in triples in whichthe outer tertiary pumps the eccentricity of the innerbinary to large values. The inner binary is subsequentlytidally dissipated into a shorter orbit (Kiseleva et al.1998). This scenario may also explain the origin of thelarge eccentricities observed in young early-type closebinaries (§9.2).We further investigate the correlation between

triples and close binaries as a function of primaryspectral type. Unfortunately, we cannot repeat theTokovinin et al. (2006) measurement for early-typesystems due to the observational selection effects and

Fig. 39.— The close companion fraction F0.3<logP<0.8;q>0.1as a function of overall triple plus quadruple star fractionFn≥2;q>0.1 colored according to primary spectral type. Forsolar-type primaries (red), F0.3<logP<0.8;q>0.1 = (1.5± 0.6)%have close companions with q > 0.1 and P ≈ 2 - 6 days, andFn≥2;q>0.1 = (10± 2)% are in triple or quadruple systems.Meanwhile, F0.3<logP<0.8;q>0.1 = (17±5)% of O-type stars havevery short-period companions and Fn≥2;q>0.1 = (73± 16)% ofO-type stars have n ≥ 2 companions with q > 0.1 (magenta). Ifdynamical evolution in triple/quadruple systems is the dominantformation mechanism of close binaries, then it must be a relativelyefficient process. For every triple/quadruple system, (16 - 22)%have inner binaries with short periods P = 2 - 6 days, irrespectiveof primary mass (dotted line).

incompleteness. We instead compare in Fig. 39 thefraction F0.3<logP<0.8;q>0.1 of primaries that have closecompanions with P = 2 - 6 days and q > 0.1 tothe overall triple/quadruple star fraction Fn≥2;q>0.1= Fn=2;q>0.1 +Fn=3;q>0.1. Although we cannotdirectly associate early-type close binaries with tertiarycompanions on a system by system basis, the correlationbetween F0.3<logP<0.8;q>0.1 and Fn≥2;q>0.1 is intriguingfor three reasons.First, F0.3<logP<0.8;q>0.1 is nearly directly proportional

to Fn≥2;q>0.1. By fitting a linear relation to the fourdata points in Fig. 39, we measure the y-intercept tobe F0.3<logP<0.8;q>0.1 = −0.005± 0.007, which is slightlysmaller than but consistent with zero. If a process otherthan dynamical evolution in triples was the dominantformation mechanism for producing close binaries, thenwe would expect the y-intercept to be measurably greaterthan zero, i.e., there would be close binaries even if therewere no triple/quadruple systems.Second, the slope of the relation ϵ =

F0.3<logP<0.8;q>0.1/Fn≥2;q>0.1 = (19± 3)% providesa direct constraint for the efficiency of close binaryformation via triple-star dynamical evolution. If not allclose binaries have outer tertiaries, then the efficiencyϵ would be correspondingly smaller. For solar-typesystems, where we know ≈70 - 90% of close binaries withP = 2 - 6 days have outer tertiaries, then the efficiencymust be ϵ ≈ 15%. This is substantially larger thanthat expected if the orbital periods of the inner andouter companions in triples were uncorrelated. Forexample, if we randomly select Pinner and Pouter fromthe underlying period distribution flogP;q>0.1 with theadded constraint that Pouter " 10Pinner for long-termdynamical stability, then only 6% of solar-type triples

F clo

se

Ftriple/quadruple

Fclose = 0.16 Ftriple/quadruple

690 A. Tokovinin et al.: Tertiary companions to SB

Fig. 13. Period distributions of TCs around close SBs (squares) andwide SBs (triangles). The error bars correspond to ±1σ. The distri-bution of solar-type binaries from DM91 is plotted for comparison(dotted line and crosses).

Table 6. Frequency of tertiary companions in two sub-samples. N isthe number of systems in each sample, fraw is the TC frequency un-corrected for incomplete detections, f is the TC frequency estimatedby the ML method. The numbers in italics are obtained by discardinguncertain companions.

Sample N fraw f

P1 < 7 d 90 0.66 0.80 ± 0.0688 0.65 0.79 ± 0.06

P1 > 7 d 75 0.33 0.40 ± 0.0671 0.30 0.36 ± 0.06

All 165 0.52 0.63 ± 0.05

SBs and field stars with respect to wide companions? We com-pare in Fig. 13 the companion frequencies in these samples,both corrected for selection effects. Overall, the companion fre-quency in the DM91 sample for periods above one yr is 50% –higher than in the wide-SB sub-sample. On the other hand, thefraction of companions to close SBs is clearly increased. Theirperiods range from a few years to several thousand years.

7.5. High multiplicities

Our sample contains 64 triples and many systems of multiplic-ity higher than 3: 11 quadruples and seven quintuples (Sect. 5).Batten (1973) introduced fn – the fraction of systems of mul-tiplicity n and higher to systems of multiplicity n − 1. For oursample, f4 = (11 + 7)/64 = 28% and f5 = 7/11 = 64%. Thesenumbers are comparable to the high-multiplicity frequencies inthe MSC: f4 = 179/626 = 29% and f5 = 38/141 = 27%(Table 1 of Tokovinin 2001). Spectroscopic binaries from theMSC, such as HIP 76563/76566 (two SBs of 3.27d and 14.28d

in a system with six components), are not included in the sam-ple. This is done to avoid intentional bias towards high multi-plicity.

Fig. 14. Frequency of tertiary companions as a function of SB period.

8. Summary

1. The period distributions of SBs with and without tertiarycompanions (TCs) are significantly different. This provesthe existence of pure SBs without TCs.

2. The mass ratio distributions of SBs with and without TCsare identical.

3. The frequency of TCs is 63% for the whole SB sample.However, it is a strong function of the SB period, reaching96% for the close (P1 < 3d) SBs and decreasing to 34%for SBs with P1 > 12d (Fig. 14). The last number is lessthan the companion frequency of ∼50% for G-dwarfs inthe field. These results are robust because the TC detectiondoes not depend on the SB period.

4. The periods of most TCs in our sample range from 2 yrto 105 yr. There is no correlation between P3 and P1. Thetriple systems with P1 < 30d have large period ratios andare very stable dynamically (Fig. 12).

5. The TCs are more massive than spectroscopic primaries ina small fraction (17% ± 4%) of multiple systems, suggest-ing that there is a tendency of the most massive componentto be found preferentially in shortest-period sub-systems.

9. Discussion

It seems that the properties of close SBs are established early intheir evolution. In a compilation of pre-Main Sequence (PMS)SBs by Melo et al. (2001), the period distribution is not differ-ent in any significant way from the distribution in our sample.For example, SB periods are divided almost equally between1d−7d and 7d−30d intervals. The observed frequency of TCs inthe Melo et al. PMS sample, 42% ± 19% (Sterzik et al. 2004),is not different from the observed TC frequency of 52% in oursample of SBs in the field.

Our main result (Fig. 14) shows that essentially all veryclose SBs are members of higher-order systems. The KCTFmechanism is most likely responsible for shortening SB peri-ods during their life on the Main Sequence. It acts indepen-dently of the TC period and mass: even a very wide and low-mass companion can influence the SB, given sufficient time.The period of Kozai cycles is of the order of P3(P3/P1) ∼106 yr for a typical tertiary period of P3 = 1000 yr and the

~80% of solar-type binaries withPinner < 7 days have tertiary companions,

while only ~30% of slightly wider binaries with Pinner > 20 days have such tertiary components

(Tokovinin+ 2006)

The very close binary fraction (P < 7 days)is directly proportional to the overall

triple/quadruple star fraction, independent of M1(Moe & Di Stefano 2017)

Very close binaries derive from dynamical interactions in triplesBUT mostly during the early pre-MS phase

Two mechanisms:

1) Kozai-Lidov oscillations in stable triples coupled with tidal friction (Kiseleva+ 98; Fabrycky & Tremaine 07)

2) Dynamical unfolding of unstable triples combined

with significant energy dissipation in disk

(Bate+ 02, 09)

Moe & Kratter 2017 (on arXiv today)

20% MS

20% pre-MS

60% pre-MS

Inclined tertiaries with aouter = 500 - 5,000 AU

Inclined tertiaries with aouter = 20 - 1,000 AU

Coplanar tertiaries with aouter = 0.5 - 50 AU

Binary Star Evolution via RLOF (Moe and Di Stefano 2017)

Frequency of companions with log P (days) < 3.7

(a < 10 AU) and q > 0.1

● Only 15% of M1 = 1 M☉ primaries will interact via RLOF● 80 - 90% of O-type primaries will experience RLOF (consistent with Sana+ 2012) ● 10 - 20% of O-type primaries are in compact triples with aouter < 10 AU

the short-period regime, we have measured the statistics ofcompanions down to logP=0.2. Despite the uncertainty in thefunctional form of the period distribution below logP<0.2,the addition of companions with logP<0.2 is negligiblecompared to the overall companion frequency at longerperiods. The frequency < < >f P q0.2 log 3.7; 0.1 of companions withmass ratios q>0.1 across orbital periods 0.2<logP<3.7 is therefore a reliable indicator of the fraction ofprimaries that experience significant binary evolutionvia RLOF.

As a function of primary mass M1, we calculate< < >f P q0.2 log 3.7; 0.1 by integrating >f P qlog ; 0.1 across 0.2<

logP<3.7. We measure the uncertainties as done in Section 9.4,and we display our results in Figure 42. Only 15%±3% of solar-type primaries experience RLOF with companions q>0.1.Meanwhile, the frequency < < >f P q0.2 log 3.7; 0.1=1.0±0.2 ofcompanions with q>0.1 and 0.2<logP<3.7 to O-typeprimaries is nearly an order of magnitude larger. Essentially allO-type primaries undergo RLOF with companions q>0.1. Infact, the measured frequency < < >f P q0.2 log 3.7; 0.1=1.0±0.2 isquite close to and may exceed unity. This suggests that ≈10%–20% of O-type primaries are in compact triple configurations inwhich the outer tertiary has q>0.1 and logPouter <3.7(aouter10 au). Close tertiaries can induce Kozai oscillationsand may cause the inner binary to merge while still on the MS,thereby producing a blue straggler (Perets & Fabrycky 2009). Ifinstead the inner binary first evolves into a pair of compactremnants, for example, the tertiary may accelerate the merger ofthe two compact objects and lead to the formation of a Type Iasupernova or short gamma-ray burst (Thompson 2011). Theevolution of compact triples should be studied in more detail,especially if they are relatively more common among massivestars.

Sana et al. (2012) report that 71% ±8% of O-type stars willinteract with companions q>0.1 via RLOF. Our estimate of

< < >f P q0.2 log 3.7; 0.1=1.0±0.2 is consistent with this estimatebut slightly larger, for two reasons. First, Sana et al. (2012)consider only binaries with P<1500 days, i.e., logP<3.2, toexperience significant binary evolution. This is primarilybecause they measure the power-law slope η=−0.4±0.2of the eccentricity distribution to be weighted toward smallvalues. Although η=−0.4 describes the eccentricity distribu-tion of short-period binaries with P<20days, we find thatmassive binaries with intermediate periods 2<logP<4 areweighted toward larger eccentricities (η≈0.8; Figure 36).Early-type binaries with slightly longer orbital periodslogP≈3.2–3.7 undergo RLOF at periastron given á ñe ≈0.5.This effect increases the fraction of O-type stars that willinteract with a companion by D>f P qlog ; 0.1 logP≈0.3×0.5=0.15 (see bottom panel of Figure 37).

Second, while Sana et al. (2012) assume that the distribu-tions of mass ratios and orbital periods are independent, we findthat early-type binaries with intermediate periods are weightedtoward smaller mass ratios. There are more companions withq≈0.1–0.4 and logP≈2–3 to O-type stars than predicted bySana et al. (2012). More recent observations with LBI confirman enhanced companion frequency at intermediate periodslogP=3.5 (Rizzuto et al. 2013; Sana et al. 2014; seeFigure 37). This second effect increases the fraction of O-typeprimaries that will interact with a binary companion by anadditional ≈15%.

Because we find that early-type binaries with intermediateorbital periods are weighted toward larger eccentricities andsmaller mass ratios, the frequency of companions that willinteract with a massive primary increases by ≈30%. We stillreaffirm the overall conclusion of Sana et al. (2012) thatmassive stars are dominated by interactions with binarycompanions. We simply find that the fraction is even larger ifwe account for the variations between P, q, and e. Moreover,the Sana et al. (2012) spectroscopic binary sample containsonly companions that are members of the inner binary.Meanwhile, LBI is sensitive to all companions q>0.3 acrossintermediate orbital periods, regardless of whether thecompanions are outer tertiaries or members of the innerbinaries. In fact, LBI surveys have detected outer tertiaries atlogPouter (days)≈3–4 to massive stars in compact tripleconfigurations (Rizzuto et al. 2013; Sana et al. 2014). This iswhy we estimate that ≈80%–90% of massive stars will interactwith a companion and ≈10%–20% of massive primaries are incompact triple configurations with logPouter (days)<3.7(aouter10au). Combining these two statistics brings thetotal close companion frequency to our measured value

< < >f P q0.2 log 3.7; 0.1=1.0±0.2 for massive O-type MSprimaries.We next utilize the measured multiplicity statistics to

estimate the fraction �evol of early-type primaries that areactually the products of binary evolution. The fraction �evolincludes not only close binaries that merge or experience stableMT but also wide companions in binaries in which the trueprimaries have already evolved into compact remnants. Using aMonte Carlo technique, we simulate a large population ofsingle and binary early-type stars (similar to our methods inSection 8.3.2 for solar-type systems). We first select primariesacross 4Me<M1<40Me from a Salpeter IMF. Given M1,we then determine the properties of the companions, i.e.,intrinsic frequency, period, and mass ratio, based on ourprobability distributions ( ∣ )f P q M, 1 measured in Section 9.Once we generate our initial population, we evolve each

binary according to the stellar evolutionary tracks of Bertelliet al. (2008, 2009) and the following assumptions regarding

Figure 42. Frequency of companions with q>0.1 and 0.2<log P(days)<3.7 per primary as a function of primary mass M1. Only 15%±3% of solar-type primaries (red) will experience significant binary evolutionvia RLOF. Meanwhile, essentially all O-type primaries (magenta) will undergoRLOF with companions q>0.1. About 10%–20% of O-type primaries are incompact triples in which the outer tertiary has log P<3.7 and may thereforesignificantly affect the evolution of the inner binary.

49


Binary Star Evolution via RLOF (Moe and Di Stefano 2017)

Even for a < 10 AU, the distribution of ZAMS binary properties are highly correlated(see also Abt+ 1990 and Duchene & Kraus 2013):

f(M1, P, q, e) ≠ f(M1) × f(P) × f(q) × f(e)

The density of binaries in certain pockets of the f(M1, P, q, e) parameter space differs by up to a factor of ~50 compared to canonical initial conditions

adopted in many binary population synthesis studies

Monte Carlo code that generates population of binariesbased on observed f(M1, P, q, e) is available

Separately adjusting the individual distributions f(P), f(q), and f(e) to theextremes will still not encompass the true nature of the binary population

Binary evolution affects your multiplicity statistics (Moe & Di Stefano 2017)

Solar-type SB1s:1) Sirius-like binaries

with hot WDs2) Barium stars

Early-type SB1s:1) EBs vs. SB1s2) N(SB1s)/N(SB2s)

increases with age

30% ± 10% of massive stars are the products of binary evolution (de Mink+ 2014)

20% ± 10% of early-type “primaries” are actually the secondaries in which the true primaries have already evolved into compact remnants

11% ± 4% of solar-type “primaries” have WD companions

For a volume-limited sample:

30% ± 10% of SB1s contain compact remnant companions

Malachi Regulus Moe

Regulus (α Leonis - the heart of the lion):- SB1 with P = 40 days- Rapidly rotating B-type star- Companion either K-dwarf or WD

30 ± 10% of SB1s have compact remnant companions (Moe & Di Stefano 2017)

?

How the close binary fraction changes withdecreasing metallicity Z

ReferenceSpectralType

Minimumlog(Z/Z☉)

As Z↓,ΔF/F

Carney+2005 G −2.4 <|30%|

Gao+ 2014 G −1.5 +50%

Hettinger+2015 F −1.7 −25%

Moe+2013 B −0.7 <|20%|

Dunstall+2015 B −0.4 <|30%|

Variations with respect to Z are small and possibly due to sensitivity and selection biases, e.g., lower-metallicity stars are systematically older and more likely to contain WD companions

Multiplicity Statistics: Diagnostics for Binary Star Formation and Initial Conditions for Binary Star Evolution (Moe & Di Stefano 2017)






log P (days) = 1 - 5;a = 0.1 - 100 AU;

Disk Fragmentation

Very Close Binarieslog P (days) < 1;

a < 0.1 AU;Dynamical Hardening in Triples during Pre-MS



• fclose = 0.02 (M1 = 1M☉) - 0.2 (30M☉)• Most have outer tertiaries• Uniform f(q) with excess twin fraction

Statistics of Binary & Multiple StarsThe Astrophysical Journal Supplement Series, 230:15 (55pp),...

Documents

Transcript of Statistics of Binary & Multiple StarsThe Astrophysical Journal Supplement Series, 230:15 (55pp),...